An Introduction to Decision Analysis, Robert T. Clemen presents an example in which he discusses the 1982 John Hinckley trial. In describing the case, Clemen says: In 1982 John Hinckley was on trial, accused of having attempted to kill President Reagan. During Hinckley’s trial, Dr. Daniel R. Weinberger told the court that when individuals diagnosed as schizophrenics were given computerized axial tomography (CAT) scans, the scans showed brain atrophy in 30% of the cases compared with only 2% of the scans done on normal people. Hinckley’s defense attorney wanted to introduce as evidence Hinckley’s CAT scan, which showed brain atrophy. The defense argued that the presence of atrophy strengthened the case that Hinckley suffered from mental illness.
a. Approximately 1.5 percent of the people in the United States suffer from schizophrenia. If we consider the prior probability of schizophrenia to be .015, use the information given to find the probability that a person has schizophrenia given that a person’s CAT scan shows brain atrophy.
b. John Hinckley’s CAT scan showed brain atrophy. Discuss whether your answer to part a helps or hurts the case that Hinckley suffered from mental illness.
c. It can be argued that .015 is not a reasonable prior probability of schizophrenia. This is because .015 is the probability that a randomly selected U. S. citizen has schizophrenia. However, John Hinckley was not a randomly selected U. S. citizen. Rather, he was accused of attempting to assassinate the President. Therefore, it might be reasonable to assess a higher prior probability of schizophrenia. Suppose you are a juror who believes there is only a 10 percent chance that Hinckley suffers from schizophrenia. Using .10 as the prior probability of schizophrenia, find the probability that a person has schizophrenia given that a person’s CAT scan shows brain atrophy.
d. If you are a juror with a prior probability of .10 that John Hinckley suffers from schizophrenia and given your answer to part c, does the fact that Hinckley’s CAT scan showed brain atrophy help the case that Hinckley suffered from mental illness?
e. If you are a juror with a prior probability of .25 that Hinckley suffers from schizophrenia, find the probability of schizophrenia given that Hinckley’s CAT scan showed brain atrophy. In this situation, how strong is the case that Hinckley suffered from mental illness?